2 MICHAEL D. FRIED, DAN HARAN, AND HELMUT VOLKLEIN variety V defined over K has a K-rational point, provided that V has a non- singular point over each real closure of K. The latter condition on Vis equivalent to the following one: the function field K(V) of V is a totally real extension of K, that is, every ordering on K extends to K(V). The field Qltr of all totally real algebraic numbers is the fixed field of all involutions in the absolute Galois group G(Ql) of Ql. From Pop [P], Qltr is PRC. By Weissauer's theorem [FJ, Proposition 12.4] every proper finite extension of Qltr is Hilbertian. Also, by Prestel's extension theorem [P2, Theorem 3.1] every algebraic extension of Qltr is PRC. Hence, the absolute Galois group of a proper finite extension of Qltr is known [FV3]. The field Qltr, however, is not Hilbertian. For example, Z 2 - ( a2 + 1) is reducible over Qltr for every a E Qltr. In [FHV] we cover also the case of Qltr. The key observation is that Qltr satisfies a certain weakening of the Hilbertian property. This allows specializing Galois extensions of K(x) whose Galois group is generated by real involutions to obtain Galois extensions of K with the same Galois group. As a result we determine the absolute Galois group of Qltr. In the present paper we extend the methods and results of [FV3]. This solves regular embedding problems over a PRC field K so that the orderings of K extend to prescribed subfields (Theorem 5.2). Thus, Theorem 5.3 gives new information about the absolute Galois group of the field K(x) of rational functions over K . The first 5 sections setup the proof of Theorem 5.2. We need an approximation theorem for varieties over PRC fields (section 1), supplements about the moduli spaces (section 2), group-theoretic lemmas (section 3), and the determination of the real involutions in Galois groups over JR(x) (section 4). In section 6 we define the concept of totally real Hilbertian, and show that Qltr has this property. Section 7 combines these results to determine the absolute Galois group of a countable totally real Hilbertian PRC field satisfying these properties: it has no proper totally real algebraic extensions and its space of orderings has no isolated points. This group is the free product of groups of order 2, indexed by the Cantor set Xw (Theorem 7.6). In particular, G(Qltr) = Aut(Q/Qltr) is isomorphic to this group. As a corollary we introduce the notion of real Frobenius fields: Qltr is an example (Corollary 8.3). Following the Galois stratification procedure of [FJ, Chap. 25] and [HL] we show that the elementary theory of real Frobenius fields allows elimination of quantifiers in the appropriate language. In particular, Qltr is primitive recursively decidable (Theorem 10.1). On the other hand the ring of integers of Qltr is undecidable (A. Prestel pointed us to Julia Robinson's proof of this [R2].) Thus we obtain a natural example of an undecidable ring with a decidable quotient field. Compare this with the possibility that Ql is decidable (cf. Robinson [Rl, p. 951]). Furthermore, we give (Corollary 10.5) a system of axioms for the theory of Qltr. Affirmations. We are grateful to Moshe Jarden for numerous suggestions
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